4.4 Trigonometric Functions of any Angle

1. 4.4 Trigonometric Functions of any Angle Objective: Students will know how to evaluate trigonometric functions of any angle, and use reference angles to evaluate trigonometric functions.

2. Let θ be an angle in standard position, with (x, y) a point on the terminal side be a point distinct from the origin on the terminal side of θ. Let r = .

3. Ex 1) The point (-4, 10) is on the terminal side of an angle in standard position. Determine the exact values of the 6 trig functions. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

4. Signs of Trig Functions

5. Ex 2) State the quadrant in which θ lies.a) tanθ >0b) sec θ<0 c) sin θ < 0 and cos θ < 0.d) cot θ > 0 and cos θ > 0.e) tan θ <0 and csc θ >0

6. Ex 3) Find the values of the 6 trig functions of θ Function ValueConstraint sin θ = 3/5 θ is in QII sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

7. Ex 4) Find the values of the 6 trig functions of θ Function ValueConstraint tan θ = -15/8 sin θ < 0 sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

8. reference angle: the acute angle formed by the terminal side of θ and the x-axis is called the reference angle

9. Ex 5) Find the reference angle θ’ for the special angle θ. Then sketch θ and θ’ in standard position. • θ = 120º • θ = -135º • θ = -5 /6 • d) θ = - /12

10. Ex 6) Evaluate the sine, cosine, and tangent without using a calculator. • θ = -750º • θ = -7 /6 sin θ = cos θ = tan θ = sin θ = cos θ = tan θ =

11. Ex 7) Find the indicated trig value in the specified quadrant. • FunctionQuadrantTrig Value • a) tan θ = 3/2 III sec θ • b) sin θ = 1/3 II cos θ

12. Example 8) Find two solutions of the equation in the first revolution. • sin θ = ½ • tan θ = -1 • csc θ =